Unit B3 · The Particulate Nature of MatterUnit B3 · 物质的微粒本质
Gas Laws气体定律
IB-Style Practice QuestionsIB 风格练习题
MEDIUMHARDPaper 1Paper 1BPaper 2HL ONLY
Syllabus B3.1 to B3.6考纲 B3.1 至 B3.6PHYSICS HL
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PART I · PAPER 1 STYLE第一部分 · 第一卷风格Short structured · calculator · 30 marks短结构题 · 可用计算器 · 30 分
Short Structured Items短结构题
Show all working in the space below each question. Marks are awarded for correct method as well as final answers. Convert every temperature to Kelvin before substituting, and work in SI units (Pa, $\mathrm{m^{3}}$, K). Give numerical answers to an appropriate number of significant figures.在每题下方空白处写出全部解题过程。方法分（method marks）与最终答案同等重要。代入前把每个温度都换成开尔文，并使用 SI 单位（Pa、$\mathrm{m^{3}}$、K）。数值答案保留适当的有效数字。
A fixed mass of gas at a pressure of $1.0\times 10^{5}\ \mathrm{Pa}$ occupies a volume of $0.024\ \mathrm{m^{3}}$. It is then compressed isothermally to a volume of $0.0060\ \mathrm{m^{3}}$.一定质量的气体在 $1.0\times 10^{5}\ \mathrm{Pa}$ 的压强下占据 $0.024\ \mathrm{m^{3}}$ 的体积，随后被等温压缩到 $0.0060\ \mathrm{m^{3}}$。
(a)State which empirical gas law applies, and the quantity that is held constant.写出适用的经验气体定律，以及保持恒定的物理量。[1]
(b)Calculate the new pressure of the gas.计算气体的新压强。[2]
(c)State what happens to the internal energy of the gas during this change, and justify your answer in one sentence.说明此过程中气体内能如何变化，并用一句话说明理由。[1]
Q3MEDIUMPaper 1ideal gas equation + Gay-Lussac理想气体方程与盖-吕萨克[6 marks]
A sealed rigid container holds gas at $1.5\times 10^{5}\ \mathrm{Pa}$ and a temperature of $300\ \mathrm{K}$.一个密封刚性容器内的气体处于 $1.5\times 10^{5}\ \mathrm{Pa}$、温度 $300\ \mathrm{K}$。
(a)The container is cooled to $250\ \mathrm{K}$. Calculate the new pressure, stating the gas law you use.容器被冷却到 $250\ \mathrm{K}$。计算新压强，并写出所用的气体定律。[3]
(b)A separate $0.50\ \mathrm{mol}$ sample of an ideal gas is held at $1.0\times 10^{5}\ \mathrm{Pa}$ and $300\ \mathrm{K}$. Calculate the volume it occupies, using $pV = nRT$.另取 $0.50\ \mathrm{mol}$ 理想气体，处于 $1.0\times 10^{5}\ \mathrm{Pa}$、$300\ \mathrm{K}$。用 $pV = nRT$ 计算其所占体积。[3]
Kinetic theory models the pressure of an ideal gas as $pV = \tfrac{1}{3} N m \overline{c^{2}}$, where $m$ is the mass of one molecule and $\overline{c^{2}}$ is the mean-square speed.气体动理论把理想气体的压强建模为 $pV = \tfrac{1}{3} N m \overline{c^{2}}$，其中 $m$ 为单个分子的质量，$\overline{c^{2}}$ 为均方速率。
(a)State two assumptions made about the molecules of an ideal gas in this model.写出该模型对理想气体分子所作的两条假设。[2]
(b)Explain, in molecular terms, why a gas exerts a pressure on the walls of its container.从分子层面解释气体为何对容器壁施加压强。[2]
(c)A gas has number density $n_V = 2.7\times 10^{25}\ \mathrm{m^{-3}}$, molecular mass $m = 4.7\times 10^{-26}\ \mathrm{kg}$ and mean-square speed $\overline{c^{2}} = 2.6\times 10^{5}\ \mathrm{m^{2}\,s^{-2}}$. Calculate the pressure using $p = \tfrac{1}{3} n_V m \overline{c^{2}}$.某气体数密度 $n_V = 2.7\times 10^{25}\ \mathrm{m^{-3}}$，分子质量 $m = 4.7\times 10^{-26}\ \mathrm{kg}$，均方速率 $\overline{c^{2}} = 2.6\times 10^{5}\ \mathrm{m^{2}\,s^{-2}}$。用 $p = \tfrac{1}{3} n_V m \overline{c^{2}}$ 计算压强。[3]
(d)State how the pressure would change if the number density alone were doubled at constant temperature.说明若仅在恒温下把数密度翻倍，压强将如何变化。[1]
A monatomic ideal gas is held at an absolute temperature of $400\ \mathrm{K}$.一种单原子理想气体保持在绝对温度 $400\ \mathrm{K}$。
(a)By equating $pV = \tfrac{1}{3} N m \overline{c^{2}}$ with $pV = N k_B T$, show that the average translational kinetic energy per molecule is $\bar{E}_k = \tfrac{3}{2} k_B T$.通过令 $pV = \tfrac{1}{3} N m \overline{c^{2}}$ 与 $pV = N k_B T$ 相等，证明每个分子的平均平动动能为 $\bar{E}_k = \tfrac{3}{2} k_B T$。[3]
(b)Calculate the average translational kinetic energy of one molecule at $400\ \mathrm{K}$.计算 $400\ \mathrm{K}$ 时一个分子的平均平动动能。[2]
(c)Calculate the internal energy of $3.0\ \mathrm{mol}$ of this gas at $400\ \mathrm{K}$, and state why for a monatomic ideal gas this is purely kinetic.计算 $3.0\ \mathrm{mol}$ 该气体在 $400\ \mathrm{K}$ 时的内能，并说明对单原子理想气体为何内能纯为动能。[3]
PART II · PAPER 1B / DATA ANALYSIS第二部分 · 第一卷 B / 数据分析Graphs · data · uncertainties · 22 marks图像 · 数据 · 不确定度 · 22 分
Graph and Data Questions图像与数据题
These items reward correct linearisation of the gas laws and careful handling of gradients and uncertainties. Quote uncertainties to one significant figure and round the value to match.这些题考查对气体定律的正确线性化，以及对斜率与不确定度的细致处理。不确定度保留 1 位有效数字，并使数值的末位与之对齐。
A fixed mass of gas is compressed at constant temperature. A student records the pressure $p$ at several volumes $V$ and tabulates $p$ against $1/V$:一定质量的气体在恒温下被压缩。学生记录若干体积 $V$ 对应的压强 $p$，并将 $p$ 对 $1/V$ 列表：
$1/V\ /\ \mathrm{m^{-3}}$
$25$
$33.3$
$50$
$100$
$p\ /\ 10^{5}\ \mathrm{Pa}$
$0.60$
$0.80$
$1.20$
$2.40$
(a)Starting from Boyle's law, show that a graph of $p$ against $1/V$ should be a straight line through the origin, and state what the gradient represents.从玻意耳定律出发，证明 $p$ 对 $1/V$ 的图应为过原点的直线，并说明斜率代表什么。[3]
(b)Calculate the gradient of the line, and hence state the value of $pV$ for this gas (the constant in Boyle's law).计算该直线的斜率，由此写出该气体的 $pV$ 值（玻意耳定律中的常数）。[3]
(c)At the highest-pressure point the pressure $2.40\times 10^{5}\ \mathrm{Pa}$ has an absolute uncertainty of $\pm 0.05\times 10^{5}\ \mathrm{Pa}$. Calculate the percentage uncertainty in this pressure reading.在压强最高的数据点处，压强 $2.40\times 10^{5}\ \mathrm{Pa}$ 的绝对不确定度为 $\pm 0.05\times 10^{5}\ \mathrm{Pa}$。计算该压强读数的百分比不确定度。[2]
(d)State one reason why, at very high pressures, the plotted points might begin to curve away from the straight line predicted by Boyle's law.说明在极高压下，所绘数据点为何可能开始偏离玻意耳定律预测的直线的一个原因。[2]
A fixed amount of an ideal gas is taken through a series of states. At each state the product $pV$ (in joules) and the absolute temperature $T$ (in kelvin) are recorded:一定量的理想气体经历一系列状态。在每个状态记录乘积 $pV$（焦耳）与绝对温度 $T$（开尔文）：
$T\ /\ \mathrm{K}$
$200$
$300$
$400$
$500$
$pV\ /\ \mathrm{J}$
$830$
$1250$
$1660$
$2080$
(a)Starting from the ideal gas equation, show that a graph of $pV$ against $T$ should be a straight line through the origin for a fixed amount of gas, and state what the gradient represents.从理想气体方程出发，证明对固定气体量，$pV$ 对 $T$ 的图应为过原点的直线，并说明斜率代表什么。[3]
(b)Using the first and last data points, calculate the gradient of the line.用首末两个数据点计算该直线的斜率。[3]
(c)Hence determine the amount of substance $n$, in moles, of the gas sample.由此求气体样品的物质的量 $n$（以摩尔计）。[2]
(d)Explain why this graph is expected to pass through the origin, and what a measured value of $pV$ at $T = 0\ \mathrm{K}$ would physically represent.解释为何该图预期过原点，以及在 $T = 0\ \mathrm{K}$ 处测得的 $pV$ 值在物理上代表什么。[2]
(e)The temperatures had been recorded in degrees Celsius instead of kelvin. State the effect this would have on the shape and intercept of the graph.若温度被记成摄氏度而非开尔文，说明这会对图像的形状与截距产生什么影响。[2]
PART III · PAPER 2 STYLE第三部分 · 第二卷风格Extended structured · calculator · 30 marks长结构题 · 可用计算器 · 30 分
Extended Structured Problems长结构问题
Set up each problem clearly, stating the equation and the variable held constant. Method marks dominate the longer items; carry intermediate values to extra figures and round only the final answer.每题先清晰列式，写明所用方程与保持恒定的变量。长题中方法分占比最大；中间值多保留几位，仅在最终答案处取舍有效数字。
Q8HARDPaper 2moles from $pV=nRT$ then heating由 $pV=nRT$ 求摩尔再加热[12 marks]
A rigid cylinder of volume $0.025\ \mathrm{m^{3}}$ contains a monatomic ideal gas at a pressure of $2.0\times 10^{5}\ \mathrm{Pa}$ and a temperature of $300\ \mathrm{K}$.一个体积 $0.025\ \mathrm{m^{3}}$ 的刚性气缸内含单原子理想气体，压强 $2.0\times 10^{5}\ \mathrm{Pa}$，温度 $300\ \mathrm{K}$。
(a)Calculate the amount of substance, in moles, of gas in the cylinder.计算气缸中气体的物质的量（以摩尔计）。[3]
(b)Determine the number of gas molecules in the cylinder.求气缸中的气体分子数。[2]
(c)The gas is heated at constant volume until its temperature reaches $500\ \mathrm{K}$. Calculate the new pressure.气体在恒容下加热至温度达 $500\ \mathrm{K}$。计算新压强。[2]
(d)Using $U = \tfrac{3}{2} n R T$, calculate the increase in the internal energy of the gas as it is heated from $300\ \mathrm{K}$ to $500\ \mathrm{K}$.用 $U = \tfrac{3}{2} n R T$ 计算气体从 $300\ \mathrm{K}$ 加热到 $500\ \mathrm{K}$ 时内能的增加量。[3]
(e)State what happens to the average translational kinetic energy of a single molecule during this heating, and by what factor it changes.说明此加热过程中单个分子的平均平动动能如何变化，以及变为原来的几倍。[2]
Q9HARDPaper 2combined gas law + real vs ideal合并气体定律与真实/理想气体[10 marks]
A fixed mass of gas in a sealed cylinder fitted with a piston is initially at $1.0\times 10^{5}\ \mathrm{Pa}$, $0.020\ \mathrm{m^{3}}$ and $300\ \mathrm{K}$. The gas is compressed and simultaneously warmed to a final state of $0.010\ \mathrm{m^{3}}$ and $350\ \mathrm{K}$.一个带活塞的密封气缸中有一定质量的气体，初态为 $1.0\times 10^{5}\ \mathrm{Pa}$、$0.020\ \mathrm{m^{3}}$、$300\ \mathrm{K}$。气体被压缩并同时加热至末态 $0.010\ \mathrm{m^{3}}$、$350\ \mathrm{K}$。
(a)State the combined gas law and the condition under which it may be applied to this gas.写出合并气体定律，以及它适用于该气体的条件。[2]
(b)Calculate the final pressure of the gas.计算气体的末压强。[3]
(c)State the four assumptions made about an ideal gas, and identify the two that are most likely to fail as this gas is strongly compressed and cooled towards condensation.写出对理想气体所作的四条假设，并指出当该气体被强烈压缩并冷却趋近凝结时，最可能失效的两条。[3]
(d)State the conditions of pressure and temperature under which a real gas behaves most like an ideal gas, and explain why in terms of the molecules.写出真实气体在何种压强与温度条件下最接近理想气体，并从分子角度解释原因。[2]
A sample of helium gas (molar mass $M = 4.0\ \mathrm{g\,mol^{-1}}$) is held at a temperature of $300\ \mathrm{K}$. Helium may be treated as a monatomic ideal gas.一份氦气（摩尔质量 $M = 4.0\ \mathrm{g\,mol^{-1}}$）保持在温度 $300\ \mathrm{K}$。氦可视为单原子理想气体。
(a)Calculate the mass of a single helium atom.计算一个氦原子的质量。[2]
(b)Calculate the average translational kinetic energy of a helium atom at $300\ \mathrm{K}$.计算 $300\ \mathrm{K}$ 时一个氦原子的平均平动动能。[2]
(c)Using $\tfrac{1}{2} m \overline{c^{2}} = \tfrac{3}{2} k_B T$, calculate the root-mean-square speed of the helium atoms.用 $\tfrac{1}{2} m \overline{c^{2}} = \tfrac{3}{2} k_B T$ 计算氦原子的方均根速率。[2]
(d)A sample of argon ($M = 40\ \mathrm{g\,mol^{-1}}$) is held at the same temperature. State, with a reason, how the rms speed of the argon atoms compares with that of the helium atoms.另取氩气（$M = 40\ \mathrm{g\,mol^{-1}}$）保持在同一温度。写出氩原子的方均根速率与氦原子相比如何，并说明理由。[2]